The report of 2014 is that academic professor Hans Freudenthal (1905-1990) committed fraud. He stole major didactic ideas from mathematics teacher Pierre van Hiele (1909-2010) with his practical experience, misrepresented those, and claimed without evidence that these falsified ways would work in the classroom.
- The implication of this report is that he was a crook.
- The report is not that Freudenthal was a demon and ate babies for breakfast.
On occasion, someone reacts that Freudenthal did also some good things. I don’t know why anyone deems it necessary to tell me this. Does someone think that I hold Freudenthal for a demon ? Really, if I had discovered that Freudenthal ate babies for breakfast, I would have reported this. None of that. The report is just what it is about.
Indeed, the implication is that major elements in mathematics education in the world have been built upon this fraud. When math education generates poor results, one element is that it doesn’t use the proper Van Hiele theory.
A reader may be so shocked by this, that it translates for brain cells A into: “this is so horrible that Freudenthal might as well be a demon”, so that brain cells B feel the need to correct with: “ho-ho, tut-tut, calm down: fraud isn’t the same as being a demon”, whence brain cells C get confused and fire the neurons: “it must be the messenger of the bad news who told us that Freudenthal was a demon, while we discovered that he just committed some fraud, and he really did some good things too”, and then the rest of the brain D comes to the conclusion: “let us respond in that manner and then go back to sleep”. Apparently it are even highly trained people whose brains work like this.
When you feel the urge to tell me that Freudenthal did also some good things: train some brain cells E to fire neurons: “hey, it generally pays to really read what a report says”.
Freudenthal’s theft went unnoticed for a long time, not because it was such a clever ploy, indeed it is petty theft, but because of Dutch society, with its respect for the highly learned professor and dwindling respect for the common teacher. It could have been discovered by interviewers Gerard Alberts and Rainer Kaenders in 2005 if they had listened to the answers (and asked a bit more). It could have been found by Sacha la Bastide – van Gemert (LB-VG) in her 2006 thesis if she had been alert – and known more about Van Hiele’s work. It could have been discovered by David Tall when he discovered the importance of the Van Hiele levels, when Van Hiele passed away in 2010 and this caused him to think about it. But Tall misread Van Hiele, and didn’t stop to wonder whether smart Van Hiele really hadn’t seen the general applicability himself, and thus Tall didn’t investigate the other publications by Van Hiele to check it up. Tall is an expert on mathematics education and should have read it long ago in Van Hiele, Structure and Insight: A Theory of Mathematics Education. Academic Press 1986.
In May 2015 the Dutch Association of Teachers of Mathematics (NVVW) celebrated its 90th anniversary. As part of the celebration the colleagues made an electronic archive of all editions of the journal Euclides. Thus a wealth of didactic material is available now, albeit still in the sink of Dutch language.
Let me use this archive to look at two articles in Euclides:
- one by Pierre and Dina van Hiele at the occasion of their Ph. D. theses of 1957, and
- one by Hans Freudenthal of 1948, to witness that he was involved with math education already back then.
For the Van Hieles, download the article from the archive. We shall have these texts below: (i) a paragraph from the archive, (ii) transformed for use for Google Translate, (iii) my translation into English, (iv) Google’s translation (with its crookedness contributing to the fun element). This seems the best way to assure the readership that there is a decent translation. Of course I invite bodies like the MAA to generate an independent translation.
Subsequently, we will see what LB-VG makes of this 1957 article.
This entry today looks at the Van Hieles only. The next entry for tomorrow is for Freudenthal. For Freudenthal’s paper I skip step (iii) since you can do the exercise as well.
Pierre and Dieke van Hiele in Euclides 1957 (iii)
It is best to start with my translation so that you as a reader know what this is all about.
To keep in mind: the purpose is to show that the Van Hieles claimed general applicability, before Hans Freudenthal and David Tall tried to take it away from them. And we don’t want to show this by a false translation but by presenting the fact. In this translation, I adopt the English preference for shorter sentences, and I insert the word “Bildung” to better express what the Van Hieles intend.
“Above, we presented a didactic approach to introducing geometry. This approach has the advantage that students experience how you can make a field of knowledge accessible for objective consideration. For such a field a requirement is that students already have command of global structures. They experience how they proceed from those to further analysis. The approach presented here for geometry namely can be used also for other fields of knowledge. Whether it will be possible to treat such a field also in mathematical manner depends upon the nature of the field. For mathematical treatment it is necessary, amongst others, that the relations do not lose their nature when they are transformed into logical relations. For students, who have participated once in this approach, it will be easier to recognize the limitations than for those students, who have been forced to accept the logical-deductive system as a ready-made given. Thus we are dealing here with a formative value (Bildung), that can be acquired by the education in the introduction into geometry.” (Translation by TC)
The confusion by Stellan Ohlsson also surfaces here. The Van Hieles move from concrete to abstract and from global to precise. While Ohlsson’s confusing terminology has from abstract to concrete.
Pierre and Dieke van Hiele in Euclides 1957 (iv)
Here is Google Translate (our fun element):
“The above-indicated way to start the geometry teaching has the advantage that the students experience, how an area of knowledge, which one global structures owned by analysis accessible to objective considerations. The here for the geometry specified path can clear are also used for other fields of knowledge. Whether there will also be possible to the field of knowledge finally mathematize, depends on the nature of the field off. Necessary for this is, after all, among other things, that the relationships are not denatured, when they are converted into logical relationships . for those who have participated once in this method are active, it will be easier to recognize the borders than for those who have had to accept given the logical deductive system as a ready-made. so we have to do here with an educational value , which can be obtained by the teaching of the start of the geometry.” (Google Translate)
La Bastide – Van Gemert’s treatment of Van Hieles 1957
LB-VG’s thesis chapter 7, Dutch original online, has this text on page 202, my translation and emphasis:
“Now the Van Hieles had thought about it themselves as well to apply the theory of levels to other subjects in mathematics education. Already in 1957 the Van Hieles indicated, in an article in Euclides about the phenomenology of education that gives an introduction to geometry, not to exclude that possibility: [quote and footnote 82]” (Translation by TC)
The curious points about this text are:
- The Van Hieles do not limit this to mathematics education only. They speak about other fields of knowledge. It is LB-VG who puts it into the box of mathematics education only.
- It isn’t “not excluded” but emphasized.
- The general claim is in the theses (ceremony July 5 1957) under supervision of Freudenthal and not just the article (October 1 1957).
- Why not quote the full paragraph ? It would show why the Van Hieles select geometry also for its ability by excellence to teach this general lesson. Pierre van Hiele’s thesis has the word “demonstration” in the title, to that the discussion of geometry is only intended to demonstrate the general applicability (in the same manner as demonstration is used in geometry itself).
LB-VG’s text is here. Below under (i) you can see the full paragraph of the quoted text.
Pierre and Dieke van Hiele in Euclides 1957 (i)
The original full paragraph in Euclides reads:
Pierre and Dieke van Hiele in Euclides 1957 (ii)
Thanks to the 90th anniversary of NVVW we can just copy & paste. We remove the paragraph marks and double spaces, i.e. replace “^p” by a space ” “, and then replace double spaces ” ” by single space ” ” again. We also remove some glitches that the OCR interpreted as dots, dashes and quotation marks, that make Dutch look even weirder. For fairness to Google we also replace the abbreviations and put the h in mathematiseren. If you want to, you can take this text and have Google translate it into your own language.
“De hiervoor aangeduide wijze om het meetkunde-onderwijs te beginnen heeft het voordeel, dat de leerlingen ervaren, hoe men een kennisgebied, waarvan men globale strukturen bezit, door analyse voor objektieve beschouwingen toegankelijk kan maken. De hier voor de meetkunde aangegeven weg kan namelijk ook voor andere kennisgebieden gebruikt worden. Of het daar ook mogelijk zal zijn het kennisveld tenslotte te mathematiseren, hangt van de aard van het veld af. Noodzakelijk daarvoor is immers onder andere, dat de relaties niet gedenatureerd worden, wanneer zij in logische relaties worden omgezet. Voor hen, die aan deze werkwijze eens aktief hebben deelgenomen, zal het gemakkelijker zijn de grenzen te herkennen dan voor hen, die het logisch deduktieve systeem als een kant en klaar gegeven hebben moeten aanvaarden. We hebben hier dus te doen met een vormende waarde, die verkregen kan worden door het onderwijs in het begin van de meetkunde.”
Part 2 is on Freudenthal 1948.